Blender rendered animation is not smooth
It sounds like you are using the BlenderPlayer to play back your render. I think that's a tool that is meant to play games made with Blender. Using that tool (from Render>Play rendered animation), I get a lag, too. BlenderPlayer can be used for a preview of what the animation looks like, but it is unable to read image data fast enough to give you a smooth animation.

To get a smooth animation, you can encode it as a video clip. The second half of this video tutorial explains how you can do that right inside Blender. Other tools can be used for this, too.

Does the frame rate affect how smooth an animation is?

Normally, animations are rendered at either 24 or 30 fps and look smooth most of the time. Smaller frame rates might make your animation look blocky, but you don't need more than that, usually. If you have a lot of movement in your scene, you can use motion blur to make it look smoother. As far as I know, movies don't usually use frame rates higher than 30 - some games use 60 fps though.